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Try to solve this problem number 35 from Singapore Mathematics Olympiad, SMO, 2018 based on Functional Equation.

Problem – Functional Equation (SMO Entrance)

Consider integers ${1,2, \ldots, 10}$. A particle is initially -at 1 . It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time?

• 82
• 81
• 80
• 79

Key Concepts

Functional Equation

Equation

But try the problem first…

Source

Challenges an Thrills – Pre – College Mathematics

Try with Hints

First hint

If you got stuck into this problem we can start taking an expected number of steps to be $g_{n}$. We need to remember at first the particle was in 1 then it will shift to the next step so for n no of position we can expressed it as n and n -1 where n = 2,3,4,……..,100.

Now try the rest…………..

Second Hint

Now let’s continue after the last hint …………

Then $g_{n+1} = \frac {1}{2} (1+g_{n} + g_{n+1} )+ \frac {1}{2}$

which implies , $g_{n+1} = g_{n} + 2$

Now we know that,$g_{2} = 1$. Then $g_{3} = 3$, $g_{4}= 5$,………………,$g_{10}=17$

$g = g_{2}+g_{3}+g_{4}+………………..+g_{10} = 1+3+…………………+17 = 81$[ Answer]